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Product (mathematics) In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called factors. For example, 21 is the product of 3 and 7 (the result of multiplication), and is the product of and (indicating that the two factors should be multiplied together).
The mixed-product property also works for the element-wise product. If A and C are matrices of the same size, B and D are matrices of the same size, then [7] ( A ⊗ B ) ∘ ( C ⊗ D ) = ( A ∘ C ) ⊗ ( B ∘ D ) . {\displaystyle (\mathbf {A} \otimes \mathbf {B} )\circ (\mathbf {C} \otimes \mathbf {D} )=(\mathbf {A} \circ \mathbf {C ...
The sum of the table entries is the product of the polynomials. Thus: ( a + b + c ) ( w + x + y + z ) = ( a w + a x + a y + a z ) + ( b w + b x + b y + b z ) + ( c w + c x + c y + c z ) . {\displaystyle {\begin{aligned}(a+b+c)(w+x+y+z)&=(aw+ax+ay+az)\\&+(bw+bx+by+bz)\\&+(cw+cx+cy+cz).\end{aligned}}}
Given two linearly independent vectors a and b, the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b, [1] and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming.
In graph theory, the Cartesian product of two graphs G and H is the graph denoted by G × H, whose vertex set is the (ordinary) Cartesian product V(G) × V(H) and such that two vertices (u,v) and (u′,v′) are adjacent in G × H, if and only if u = u′ and v is adjacent with v′ in H, or v = v′ and u is adjacent with u′ in G.
The product rule can be generalized to products of more than two factors. For example, for three factors we have d ( u v w ) d x = d u d x v w + u d v d x w + u v d w d x . {\displaystyle {\frac {d(uvw)}{dx}}={\frac {du}{dx}}vw+u{\frac {dv}{dx}}w+uv{\frac {dw}{dx}}.}
In mathematics, a geometric algebra (also known as a Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors.
In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector. If the two coordinate vectors have dimensions n and m, then their outer product is an n × m matrix.
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used.
The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.